We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved in [8] for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.

Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses co-exist?We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a structure of gradient flow.Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation.Our motivation comes from population adaptive evolution a branch of mathematical ecology which models darwinian evolution.

Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction-diffusion equation, we explain the observed phenomena of front acceleration (when the motility is unbounded) as well as other quantitative results, such as the selection of the most motile individuals (when the motility is bounded). The key argument for the construction and analysis of traveling fronts is the derivation of the dispersion relation linking the speed of the wave and the spatial decay. When the motility is unbounded we show that the position of the front scales as t 3/2 . When the mutation rate is low we show that the canonical equation for the dynamics of the fittest trait should be stated as a PDE in our context. It turns out to be a type of Burgers equation with source term. RésuméFronts d'invasion avec motilité variable : répartition des phénotypes et accélération de l'onde. Les fronts d'invasion enécologie ontété largementétudiés. Cependant peu de résultats mathématiques existent pour le cas d'un coefficient de motilité variable (à cause des mutations). A partir d'un modèle minimal de réactiondiffusion, nous expliquons le phénomène observé d'accélération du front (lorsque la motilité n'est pas bornée), et nous démontrons l'existence d'ondes progressives ainsi que la sélection des individus les plus motiles (lorsque la motilité est bornée). Le point clé pour la construction des fronts est la relation de dispersion qui relie la vitesse de l'onde avec la décroissance en espace. Lorsque la motilité n'est pas bornée nous montrons que la position du front suit une loi d'échelle en t 3/2 . Lorsque le taux de mutation est faible, nous montrons que, dans notre contexte, l'équation canonique pour la dynamique du meilleur trait est une EDP. C'est uneéquation de type Burgers avec terme source.

Adaptation in spatially heterogeneous environments results from the balance between local selection, mutation, and migration. We study the interplay among these different evolutionary forces and demography in a classical two-habitat scenario with asexual reproduction. We develop a new theoretical approach that goes beyond the Adaptive Dynamics framework, and allows us to explore the effect of high mutation rates on the stationary phenotypic distribution. We show that this approach improves the classical Gaussian approximation, and captures accurately the shape of this equilibrium phenotypic distribution in one- and two-population scenarios. We examine the evolutionary equilibrium under general conditions where demography and selection may be nonsymmetric between the two habitats. In particular, we show how migration may increase differentiation in a source–sink scenario. We discuss the implications of these analytic results for the adaptation of organisms with large mutation rates, such as RNA viruses.

We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP equation. This rescaling is based on the exponential speed of propagation of the population. In particular we show that the only role of the fractional Laplacian in determining this speed is at the initial layer where it determines the thickness of the tails of the solutions. Next, we show that such rescaling is also possible for models with non-local reaction terms, as selection-mutation models. However, to obtain a more relevant qualitative behavior for this second case, we introduce, in the second part of the paper, a second rescaling where we assume that the diffusion steps are small. In this way, using a WKB ansatz, we obtain a Hamilton-Jacobi equation in the limit which describes the asymptotic dynamics of the solutions, similarly to the case of selection-mutation models with a classical Laplace term or an integral kernel with thin tails. However, the rescaling introduced here is very different from the latter cases. We extend these results to the multidimensional case.

We study sexual populations structured by a phenotypic trait and a space variable, in a non-homogeneous environment. Departing from a structured population equation we perform a hydrodynamic-type limit to derive a model close to an existing model of theoretical biology. We then perform a further simplification to obtain a model depending on only one parameter that indicates how fast the environment is changing. We show that depending on this parameter, there exist either propagating waves, where the population invades the entire environment, or steady-states where the population survives but remains in a limited range. The corresponding propagating fronts connect an unstable steady point to a singular point. Existence of steady states with limited range distinguishes the dynamics of the sexual populations from asexual populations, where the populations whether gets extinct or propagates to the whole environment. Numerical simulations show that the derived simplified model is a good approximation of the initial structured population model.

We study a non-local parabolic Lotka-Volterra type equation describing a population structured by a space variable x ∈ R d and a phenotypical trait θ ∈ Θ. Considering diffusion, mutations and space-local competition between the individuals, we analyze the asymptotic (longtime/long-range in the x variable) exponential behavior of the solutions. Using some kind of real phase WKB ansatz, we prove that the propagation of the population in space can be described by a Hamilton-Jacobi equation with obstacle which is independent of θ. The effective Hamiltonian is derived from an eigenvalue problem. The main difficulties are the lack of regularity estimates in the space variable, and the lack of comparison principle due to the non-local term.

We study the long time behavior of a parabolic Lotka-Volterra type equation considering a time-periodic growth rate with non-local competition. Such equation describes the dynamics of a phenotypically structured population under the effect of mutations and selection in a fluctuating environment. We first prove that, in long time, the solution converges to the unique periodic solution of the problem. Next, we describe this periodic solution asymptotically as the effect of the mutations vanish. Using a theory based on Hamilton-Jacobi equations with constraint, we prove that, as the effect of the mutations vanishes, the solution concentrates on a single Dirac mass, while the size of the population varies periodically in time. When the effect of the mutations is small but nonzero, we provide some formal approximations of the moments of the population's distribution. We then show, via some examples, how such results could be compared to biological experiments.